Properties

Label 1288.2.a.n.1.4
Level $1288$
Weight $2$
Character 1288.1
Self dual yes
Analytic conductor $10.285$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1288,2,Mod(1,1288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1288, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1288.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1288 = 2^{3} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1288.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.2847317803\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.8468.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 3x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.27841\) of defining polynomial
Character \(\chi\) \(=\) 1288.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.27841 q^{3} -0.477194 q^{5} +1.00000 q^{7} -1.36566 q^{9} +O(q^{10})\) \(q+1.27841 q^{3} -0.477194 q^{5} +1.00000 q^{7} -1.36566 q^{9} -2.43556 q^{11} -7.18356 q^{13} -0.610051 q^{15} +0.302704 q^{17} -5.60244 q^{19} +1.27841 q^{21} -1.00000 q^{23} -4.77229 q^{25} -5.58112 q^{27} +0.0166785 q^{29} +7.66075 q^{31} -3.11365 q^{33} -0.477194 q^{35} -8.93916 q^{37} -9.18356 q^{39} +10.6115 q^{41} +5.03039 q^{43} +0.651684 q^{45} -3.27841 q^{47} +1.00000 q^{49} +0.386981 q^{51} +3.11365 q^{53} +1.16223 q^{55} -7.16223 q^{57} -7.90514 q^{59} +10.4620 q^{61} -1.36566 q^{63} +3.42795 q^{65} -2.78454 q^{67} -1.27841 q^{69} -0.641953 q^{71} +10.3124 q^{73} -6.10096 q^{75} -2.43556 q^{77} +2.43556 q^{79} -3.03800 q^{81} -11.7343 q^{83} -0.144448 q^{85} +0.0213220 q^{87} -7.59085 q^{89} -7.18356 q^{91} +9.79361 q^{93} +2.67345 q^{95} -1.12525 q^{97} +3.32615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{3} + 4 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{3} + 4 q^{7} + q^{9} - 10 q^{11} - 3 q^{13} - 6 q^{15} - 4 q^{17} - 10 q^{19} - 3 q^{21} - 4 q^{23} + 4 q^{25} - 9 q^{27} - 13 q^{29} + 3 q^{31} + 20 q^{33} - 11 q^{39} + q^{41} - 8 q^{43} + 4 q^{45} - 5 q^{47} + 4 q^{49} - 4 q^{51} - 20 q^{53} - 22 q^{55} - 2 q^{57} - 14 q^{59} + 8 q^{61} + q^{63} - 2 q^{65} - 18 q^{67} + 3 q^{69} - 25 q^{71} + 15 q^{73} - 11 q^{75} - 10 q^{77} + 10 q^{79} - 36 q^{83} - 28 q^{85} + q^{87} + 4 q^{89} - 3 q^{91} + 17 q^{93} - 36 q^{95} + 6 q^{97} - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.27841 0.738092 0.369046 0.929411i \(-0.379684\pi\)
0.369046 + 0.929411i \(0.379684\pi\)
\(4\) 0 0
\(5\) −0.477194 −0.213408 −0.106704 0.994291i \(-0.534030\pi\)
−0.106704 + 0.994291i \(0.534030\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −1.36566 −0.455220
\(10\) 0 0
\(11\) −2.43556 −0.734349 −0.367175 0.930152i \(-0.619675\pi\)
−0.367175 + 0.930152i \(0.619675\pi\)
\(12\) 0 0
\(13\) −7.18356 −1.99236 −0.996180 0.0873224i \(-0.972169\pi\)
−0.996180 + 0.0873224i \(0.972169\pi\)
\(14\) 0 0
\(15\) −0.610051 −0.157515
\(16\) 0 0
\(17\) 0.302704 0.0734165 0.0367082 0.999326i \(-0.488313\pi\)
0.0367082 + 0.999326i \(0.488313\pi\)
\(18\) 0 0
\(19\) −5.60244 −1.28529 −0.642644 0.766165i \(-0.722163\pi\)
−0.642644 + 0.766165i \(0.722163\pi\)
\(20\) 0 0
\(21\) 1.27841 0.278973
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −4.77229 −0.954457
\(26\) 0 0
\(27\) −5.58112 −1.07409
\(28\) 0 0
\(29\) 0.0166785 0.00309712 0.00154856 0.999999i \(-0.499507\pi\)
0.00154856 + 0.999999i \(0.499507\pi\)
\(30\) 0 0
\(31\) 7.66075 1.37591 0.687956 0.725753i \(-0.258508\pi\)
0.687956 + 0.725753i \(0.258508\pi\)
\(32\) 0 0
\(33\) −3.11365 −0.542018
\(34\) 0 0
\(35\) −0.477194 −0.0806605
\(36\) 0 0
\(37\) −8.93916 −1.46959 −0.734795 0.678289i \(-0.762722\pi\)
−0.734795 + 0.678289i \(0.762722\pi\)
\(38\) 0 0
\(39\) −9.18356 −1.47055
\(40\) 0 0
\(41\) 10.6115 1.65724 0.828619 0.559812i \(-0.189127\pi\)
0.828619 + 0.559812i \(0.189127\pi\)
\(42\) 0 0
\(43\) 5.03039 0.767127 0.383564 0.923514i \(-0.374697\pi\)
0.383564 + 0.923514i \(0.374697\pi\)
\(44\) 0 0
\(45\) 0.651684 0.0971473
\(46\) 0 0
\(47\) −3.27841 −0.478206 −0.239103 0.970994i \(-0.576853\pi\)
−0.239103 + 0.970994i \(0.576853\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0.386981 0.0541881
\(52\) 0 0
\(53\) 3.11365 0.427693 0.213847 0.976867i \(-0.431401\pi\)
0.213847 + 0.976867i \(0.431401\pi\)
\(54\) 0 0
\(55\) 1.16223 0.156716
\(56\) 0 0
\(57\) −7.16223 −0.948661
\(58\) 0 0
\(59\) −7.90514 −1.02916 −0.514581 0.857442i \(-0.672053\pi\)
−0.514581 + 0.857442i \(0.672053\pi\)
\(60\) 0 0
\(61\) 10.4620 1.33952 0.669759 0.742579i \(-0.266397\pi\)
0.669759 + 0.742579i \(0.266397\pi\)
\(62\) 0 0
\(63\) −1.36566 −0.172057
\(64\) 0 0
\(65\) 3.42795 0.425185
\(66\) 0 0
\(67\) −2.78454 −0.340186 −0.170093 0.985428i \(-0.554407\pi\)
−0.170093 + 0.985428i \(0.554407\pi\)
\(68\) 0 0
\(69\) −1.27841 −0.153903
\(70\) 0 0
\(71\) −0.641953 −0.0761858 −0.0380929 0.999274i \(-0.512128\pi\)
−0.0380929 + 0.999274i \(0.512128\pi\)
\(72\) 0 0
\(73\) 10.3124 1.20698 0.603490 0.797371i \(-0.293776\pi\)
0.603490 + 0.797371i \(0.293776\pi\)
\(74\) 0 0
\(75\) −6.10096 −0.704478
\(76\) 0 0
\(77\) −2.43556 −0.277558
\(78\) 0 0
\(79\) 2.43556 0.274022 0.137011 0.990570i \(-0.456250\pi\)
0.137011 + 0.990570i \(0.456250\pi\)
\(80\) 0 0
\(81\) −3.03800 −0.337556
\(82\) 0 0
\(83\) −11.7343 −1.28801 −0.644003 0.765023i \(-0.722727\pi\)
−0.644003 + 0.765023i \(0.722727\pi\)
\(84\) 0 0
\(85\) −0.144448 −0.0156676
\(86\) 0 0
\(87\) 0.0213220 0.00228596
\(88\) 0 0
\(89\) −7.59085 −0.804628 −0.402314 0.915502i \(-0.631794\pi\)
−0.402314 + 0.915502i \(0.631794\pi\)
\(90\) 0 0
\(91\) −7.18356 −0.753041
\(92\) 0 0
\(93\) 9.79361 1.01555
\(94\) 0 0
\(95\) 2.67345 0.274290
\(96\) 0 0
\(97\) −1.12525 −0.114251 −0.0571257 0.998367i \(-0.518194\pi\)
−0.0571257 + 0.998367i \(0.518194\pi\)
\(98\) 0 0
\(99\) 3.32615 0.334290
\(100\) 0 0
\(101\) −9.46131 −0.941435 −0.470718 0.882284i \(-0.656005\pi\)
−0.470718 + 0.882284i \(0.656005\pi\)
\(102\) 0 0
\(103\) 0.348980 0.0343860 0.0171930 0.999852i \(-0.494527\pi\)
0.0171930 + 0.999852i \(0.494527\pi\)
\(104\) 0 0
\(105\) −0.610051 −0.0595349
\(106\) 0 0
\(107\) −0.223070 −0.0215650 −0.0107825 0.999942i \(-0.503432\pi\)
−0.0107825 + 0.999942i \(0.503432\pi\)
\(108\) 0 0
\(109\) 16.5264 1.58294 0.791470 0.611208i \(-0.209316\pi\)
0.791470 + 0.611208i \(0.209316\pi\)
\(110\) 0 0
\(111\) −11.4279 −1.08469
\(112\) 0 0
\(113\) −16.0181 −1.50686 −0.753430 0.657529i \(-0.771602\pi\)
−0.753430 + 0.657529i \(0.771602\pi\)
\(114\) 0 0
\(115\) 0.477194 0.0444986
\(116\) 0 0
\(117\) 9.81029 0.906961
\(118\) 0 0
\(119\) 0.302704 0.0277488
\(120\) 0 0
\(121\) −5.06804 −0.460731
\(122\) 0 0
\(123\) 13.5659 1.22320
\(124\) 0 0
\(125\) 4.66328 0.417096
\(126\) 0 0
\(127\) −0.0699024 −0.00620284 −0.00310142 0.999995i \(-0.500987\pi\)
−0.00310142 + 0.999995i \(0.500987\pi\)
\(128\) 0 0
\(129\) 6.43092 0.566211
\(130\) 0 0
\(131\) −8.14954 −0.712028 −0.356014 0.934481i \(-0.615865\pi\)
−0.356014 + 0.934481i \(0.615865\pi\)
\(132\) 0 0
\(133\) −5.60244 −0.485793
\(134\) 0 0
\(135\) 2.66328 0.229218
\(136\) 0 0
\(137\) −9.42795 −0.805484 −0.402742 0.915314i \(-0.631943\pi\)
−0.402742 + 0.915314i \(0.631943\pi\)
\(138\) 0 0
\(139\) −13.5209 −1.14683 −0.573416 0.819264i \(-0.694382\pi\)
−0.573416 + 0.819264i \(0.694382\pi\)
\(140\) 0 0
\(141\) −4.19117 −0.352960
\(142\) 0 0
\(143\) 17.4960 1.46309
\(144\) 0 0
\(145\) −0.00795888 −0.000660949 0
\(146\) 0 0
\(147\) 1.27841 0.105442
\(148\) 0 0
\(149\) 14.4157 1.18098 0.590490 0.807045i \(-0.298935\pi\)
0.590490 + 0.807045i \(0.298935\pi\)
\(150\) 0 0
\(151\) 0.418883 0.0340882 0.0170441 0.999855i \(-0.494574\pi\)
0.0170441 + 0.999855i \(0.494574\pi\)
\(152\) 0 0
\(153\) −0.413390 −0.0334206
\(154\) 0 0
\(155\) −3.65566 −0.293630
\(156\) 0 0
\(157\) 24.3555 1.94378 0.971891 0.235431i \(-0.0756500\pi\)
0.971891 + 0.235431i \(0.0756500\pi\)
\(158\) 0 0
\(159\) 3.98054 0.315677
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 4.18653 0.327914 0.163957 0.986467i \(-0.447574\pi\)
0.163957 + 0.986467i \(0.447574\pi\)
\(164\) 0 0
\(165\) 1.48582 0.115671
\(166\) 0 0
\(167\) −13.5756 −1.05051 −0.525257 0.850944i \(-0.676031\pi\)
−0.525257 + 0.850944i \(0.676031\pi\)
\(168\) 0 0
\(169\) 38.6035 2.96950
\(170\) 0 0
\(171\) 7.65102 0.585088
\(172\) 0 0
\(173\) 15.5293 1.18067 0.590337 0.807157i \(-0.298995\pi\)
0.590337 + 0.807157i \(0.298995\pi\)
\(174\) 0 0
\(175\) −4.77229 −0.360751
\(176\) 0 0
\(177\) −10.1060 −0.759617
\(178\) 0 0
\(179\) −20.5550 −1.53635 −0.768175 0.640240i \(-0.778835\pi\)
−0.768175 + 0.640240i \(0.778835\pi\)
\(180\) 0 0
\(181\) −6.06441 −0.450764 −0.225382 0.974270i \(-0.572363\pi\)
−0.225382 + 0.974270i \(0.572363\pi\)
\(182\) 0 0
\(183\) 13.3747 0.988688
\(184\) 0 0
\(185\) 4.26571 0.313622
\(186\) 0 0
\(187\) −0.737254 −0.0539133
\(188\) 0 0
\(189\) −5.58112 −0.405967
\(190\) 0 0
\(191\) −14.2838 −1.03354 −0.516771 0.856123i \(-0.672866\pi\)
−0.516771 + 0.856123i \(0.672866\pi\)
\(192\) 0 0
\(193\) 12.4037 0.892835 0.446417 0.894825i \(-0.352700\pi\)
0.446417 + 0.894825i \(0.352700\pi\)
\(194\) 0 0
\(195\) 4.38234 0.313826
\(196\) 0 0
\(197\) −10.6420 −0.758208 −0.379104 0.925354i \(-0.623768\pi\)
−0.379104 + 0.925354i \(0.623768\pi\)
\(198\) 0 0
\(199\) −11.1318 −0.789112 −0.394556 0.918872i \(-0.629102\pi\)
−0.394556 + 0.918872i \(0.629102\pi\)
\(200\) 0 0
\(201\) −3.55980 −0.251089
\(202\) 0 0
\(203\) 0.0166785 0.00117060
\(204\) 0 0
\(205\) −5.06375 −0.353667
\(206\) 0 0
\(207\) 1.36566 0.0949198
\(208\) 0 0
\(209\) 13.6451 0.943850
\(210\) 0 0
\(211\) −25.3489 −1.74509 −0.872546 0.488532i \(-0.837532\pi\)
−0.872546 + 0.488532i \(0.837532\pi\)
\(212\) 0 0
\(213\) −0.820682 −0.0562322
\(214\) 0 0
\(215\) −2.40047 −0.163711
\(216\) 0 0
\(217\) 7.66075 0.520046
\(218\) 0 0
\(219\) 13.1836 0.890862
\(220\) 0 0
\(221\) −2.17449 −0.146272
\(222\) 0 0
\(223\) 3.45900 0.231632 0.115816 0.993271i \(-0.463052\pi\)
0.115816 + 0.993271i \(0.463052\pi\)
\(224\) 0 0
\(225\) 6.51731 0.434488
\(226\) 0 0
\(227\) −4.24253 −0.281587 −0.140793 0.990039i \(-0.544965\pi\)
−0.140793 + 0.990039i \(0.544965\pi\)
\(228\) 0 0
\(229\) −14.0978 −0.931607 −0.465803 0.884888i \(-0.654235\pi\)
−0.465803 + 0.884888i \(0.654235\pi\)
\(230\) 0 0
\(231\) −3.11365 −0.204863
\(232\) 0 0
\(233\) −29.0239 −1.90142 −0.950709 0.310085i \(-0.899643\pi\)
−0.950709 + 0.310085i \(0.899643\pi\)
\(234\) 0 0
\(235\) 1.56444 0.102053
\(236\) 0 0
\(237\) 3.11365 0.202254
\(238\) 0 0
\(239\) −19.5507 −1.26463 −0.632314 0.774712i \(-0.717895\pi\)
−0.632314 + 0.774712i \(0.717895\pi\)
\(240\) 0 0
\(241\) 10.6031 0.683006 0.341503 0.939881i \(-0.389064\pi\)
0.341503 + 0.939881i \(0.389064\pi\)
\(242\) 0 0
\(243\) 12.8595 0.824939
\(244\) 0 0
\(245\) −0.477194 −0.0304868
\(246\) 0 0
\(247\) 40.2454 2.56076
\(248\) 0 0
\(249\) −15.0013 −0.950667
\(250\) 0 0
\(251\) 30.4765 1.92366 0.961829 0.273651i \(-0.0882315\pi\)
0.961829 + 0.273651i \(0.0882315\pi\)
\(252\) 0 0
\(253\) 2.43556 0.153122
\(254\) 0 0
\(255\) −0.184665 −0.0115642
\(256\) 0 0
\(257\) 13.6339 0.850462 0.425231 0.905085i \(-0.360193\pi\)
0.425231 + 0.905085i \(0.360193\pi\)
\(258\) 0 0
\(259\) −8.93916 −0.555453
\(260\) 0 0
\(261\) −0.0227771 −0.00140987
\(262\) 0 0
\(263\) −9.64044 −0.594455 −0.297227 0.954807i \(-0.596062\pi\)
−0.297227 + 0.954807i \(0.596062\pi\)
\(264\) 0 0
\(265\) −1.48582 −0.0912730
\(266\) 0 0
\(267\) −9.70424 −0.593890
\(268\) 0 0
\(269\) 25.6766 1.56553 0.782764 0.622318i \(-0.213809\pi\)
0.782764 + 0.622318i \(0.213809\pi\)
\(270\) 0 0
\(271\) 20.4801 1.24408 0.622039 0.782986i \(-0.286305\pi\)
0.622039 + 0.782986i \(0.286305\pi\)
\(272\) 0 0
\(273\) −9.18356 −0.555814
\(274\) 0 0
\(275\) 11.6232 0.700905
\(276\) 0 0
\(277\) 1.65247 0.0992876 0.0496438 0.998767i \(-0.484191\pi\)
0.0496438 + 0.998767i \(0.484191\pi\)
\(278\) 0 0
\(279\) −10.4620 −0.626342
\(280\) 0 0
\(281\) −5.35195 −0.319270 −0.159635 0.987176i \(-0.551032\pi\)
−0.159635 + 0.987176i \(0.551032\pi\)
\(282\) 0 0
\(283\) −0.731317 −0.0434723 −0.0217362 0.999764i \(-0.506919\pi\)
−0.0217362 + 0.999764i \(0.506919\pi\)
\(284\) 0 0
\(285\) 3.41777 0.202451
\(286\) 0 0
\(287\) 10.6115 0.626377
\(288\) 0 0
\(289\) −16.9084 −0.994610
\(290\) 0 0
\(291\) −1.43853 −0.0843281
\(292\) 0 0
\(293\) −10.2528 −0.598975 −0.299487 0.954100i \(-0.596816\pi\)
−0.299487 + 0.954100i \(0.596816\pi\)
\(294\) 0 0
\(295\) 3.77229 0.219631
\(296\) 0 0
\(297\) 13.5932 0.788755
\(298\) 0 0
\(299\) 7.18356 0.415436
\(300\) 0 0
\(301\) 5.03039 0.289947
\(302\) 0 0
\(303\) −12.0955 −0.694866
\(304\) 0 0
\(305\) −4.99239 −0.285863
\(306\) 0 0
\(307\) −16.9841 −0.969335 −0.484667 0.874699i \(-0.661059\pi\)
−0.484667 + 0.874699i \(0.661059\pi\)
\(308\) 0 0
\(309\) 0.446141 0.0253801
\(310\) 0 0
\(311\) 8.05831 0.456945 0.228472 0.973550i \(-0.426627\pi\)
0.228472 + 0.973550i \(0.426627\pi\)
\(312\) 0 0
\(313\) −3.68801 −0.208459 −0.104229 0.994553i \(-0.533238\pi\)
−0.104229 + 0.994553i \(0.533238\pi\)
\(314\) 0 0
\(315\) 0.651684 0.0367182
\(316\) 0 0
\(317\) −4.30965 −0.242054 −0.121027 0.992649i \(-0.538619\pi\)
−0.121027 + 0.992649i \(0.538619\pi\)
\(318\) 0 0
\(319\) −0.0406215 −0.00227437
\(320\) 0 0
\(321\) −0.285176 −0.0159170
\(322\) 0 0
\(323\) −1.69588 −0.0943613
\(324\) 0 0
\(325\) 34.2820 1.90162
\(326\) 0 0
\(327\) 21.1275 1.16836
\(328\) 0 0
\(329\) −3.27841 −0.180745
\(330\) 0 0
\(331\) 27.0285 1.48562 0.742811 0.669501i \(-0.233492\pi\)
0.742811 + 0.669501i \(0.233492\pi\)
\(332\) 0 0
\(333\) 12.2078 0.668986
\(334\) 0 0
\(335\) 1.32877 0.0725982
\(336\) 0 0
\(337\) −9.50395 −0.517713 −0.258857 0.965916i \(-0.583346\pi\)
−0.258857 + 0.965916i \(0.583346\pi\)
\(338\) 0 0
\(339\) −20.4778 −1.11220
\(340\) 0 0
\(341\) −18.6582 −1.01040
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0.610051 0.0328440
\(346\) 0 0
\(347\) −19.8863 −1.06755 −0.533776 0.845626i \(-0.679227\pi\)
−0.533776 + 0.845626i \(0.679227\pi\)
\(348\) 0 0
\(349\) 1.91064 0.102274 0.0511370 0.998692i \(-0.483715\pi\)
0.0511370 + 0.998692i \(0.483715\pi\)
\(350\) 0 0
\(351\) 40.0923 2.13997
\(352\) 0 0
\(353\) 25.4109 1.35248 0.676242 0.736680i \(-0.263607\pi\)
0.676242 + 0.736680i \(0.263607\pi\)
\(354\) 0 0
\(355\) 0.306336 0.0162586
\(356\) 0 0
\(357\) 0.386981 0.0204812
\(358\) 0 0
\(359\) 33.9497 1.79180 0.895898 0.444260i \(-0.146533\pi\)
0.895898 + 0.444260i \(0.146533\pi\)
\(360\) 0 0
\(361\) 12.3873 0.651965
\(362\) 0 0
\(363\) −6.47905 −0.340062
\(364\) 0 0
\(365\) −4.92103 −0.257579
\(366\) 0 0
\(367\) 35.2577 1.84044 0.920218 0.391405i \(-0.128011\pi\)
0.920218 + 0.391405i \(0.128011\pi\)
\(368\) 0 0
\(369\) −14.4917 −0.754408
\(370\) 0 0
\(371\) 3.11365 0.161653
\(372\) 0 0
\(373\) 21.4381 1.11002 0.555012 0.831842i \(-0.312714\pi\)
0.555012 + 0.831842i \(0.312714\pi\)
\(374\) 0 0
\(375\) 5.96159 0.307855
\(376\) 0 0
\(377\) −0.119811 −0.00617058
\(378\) 0 0
\(379\) −14.5754 −0.748686 −0.374343 0.927290i \(-0.622132\pi\)
−0.374343 + 0.927290i \(0.622132\pi\)
\(380\) 0 0
\(381\) −0.0893642 −0.00457827
\(382\) 0 0
\(383\) 28.9999 1.48183 0.740914 0.671600i \(-0.234393\pi\)
0.740914 + 0.671600i \(0.234393\pi\)
\(384\) 0 0
\(385\) 1.16223 0.0592330
\(386\) 0 0
\(387\) −6.86979 −0.349211
\(388\) 0 0
\(389\) 4.72336 0.239484 0.119742 0.992805i \(-0.461793\pi\)
0.119742 + 0.992805i \(0.461793\pi\)
\(390\) 0 0
\(391\) −0.302704 −0.0153084
\(392\) 0 0
\(393\) −10.4185 −0.525543
\(394\) 0 0
\(395\) −1.16223 −0.0584784
\(396\) 0 0
\(397\) −32.2972 −1.62095 −0.810475 0.585773i \(-0.800791\pi\)
−0.810475 + 0.585773i \(0.800791\pi\)
\(398\) 0 0
\(399\) −7.16223 −0.358560
\(400\) 0 0
\(401\) 22.6054 1.12886 0.564430 0.825481i \(-0.309096\pi\)
0.564430 + 0.825481i \(0.309096\pi\)
\(402\) 0 0
\(403\) −55.0314 −2.74131
\(404\) 0 0
\(405\) 1.44972 0.0720369
\(406\) 0 0
\(407\) 21.7719 1.07919
\(408\) 0 0
\(409\) −10.2097 −0.504838 −0.252419 0.967618i \(-0.581226\pi\)
−0.252419 + 0.967618i \(0.581226\pi\)
\(410\) 0 0
\(411\) −12.0528 −0.594522
\(412\) 0 0
\(413\) −7.90514 −0.388987
\(414\) 0 0
\(415\) 5.59953 0.274870
\(416\) 0 0
\(417\) −17.2854 −0.846468
\(418\) 0 0
\(419\) 18.2273 0.890462 0.445231 0.895416i \(-0.353122\pi\)
0.445231 + 0.895416i \(0.353122\pi\)
\(420\) 0 0
\(421\) −3.15630 −0.153829 −0.0769143 0.997038i \(-0.524507\pi\)
−0.0769143 + 0.997038i \(0.524507\pi\)
\(422\) 0 0
\(423\) 4.47719 0.217689
\(424\) 0 0
\(425\) −1.44459 −0.0700729
\(426\) 0 0
\(427\) 10.4620 0.506290
\(428\) 0 0
\(429\) 22.3671 1.07989
\(430\) 0 0
\(431\) −5.39326 −0.259784 −0.129892 0.991528i \(-0.541463\pi\)
−0.129892 + 0.991528i \(0.541463\pi\)
\(432\) 0 0
\(433\) 6.39526 0.307336 0.153668 0.988123i \(-0.450891\pi\)
0.153668 + 0.988123i \(0.450891\pi\)
\(434\) 0 0
\(435\) −0.0101747 −0.000487842 0
\(436\) 0 0
\(437\) 5.60244 0.268001
\(438\) 0 0
\(439\) −37.1943 −1.77519 −0.887594 0.460626i \(-0.847625\pi\)
−0.887594 + 0.460626i \(0.847625\pi\)
\(440\) 0 0
\(441\) −1.36566 −0.0650314
\(442\) 0 0
\(443\) −36.2228 −1.72100 −0.860498 0.509453i \(-0.829848\pi\)
−0.860498 + 0.509453i \(0.829848\pi\)
\(444\) 0 0
\(445\) 3.62231 0.171714
\(446\) 0 0
\(447\) 18.4292 0.871672
\(448\) 0 0
\(449\) −30.8859 −1.45760 −0.728799 0.684728i \(-0.759921\pi\)
−0.728799 + 0.684728i \(0.759921\pi\)
\(450\) 0 0
\(451\) −25.8450 −1.21699
\(452\) 0 0
\(453\) 0.535505 0.0251602
\(454\) 0 0
\(455\) 3.42795 0.160705
\(456\) 0 0
\(457\) −14.2852 −0.668232 −0.334116 0.942532i \(-0.608438\pi\)
−0.334116 + 0.942532i \(0.608438\pi\)
\(458\) 0 0
\(459\) −1.68943 −0.0788556
\(460\) 0 0
\(461\) −9.40069 −0.437834 −0.218917 0.975744i \(-0.570252\pi\)
−0.218917 + 0.975744i \(0.570252\pi\)
\(462\) 0 0
\(463\) 21.9015 1.01785 0.508925 0.860811i \(-0.330043\pi\)
0.508925 + 0.860811i \(0.330043\pi\)
\(464\) 0 0
\(465\) −4.67345 −0.216726
\(466\) 0 0
\(467\) 29.9877 1.38766 0.693832 0.720137i \(-0.255921\pi\)
0.693832 + 0.720137i \(0.255921\pi\)
\(468\) 0 0
\(469\) −2.78454 −0.128578
\(470\) 0 0
\(471\) 31.1364 1.43469
\(472\) 0 0
\(473\) −12.2518 −0.563339
\(474\) 0 0
\(475\) 26.7364 1.22675
\(476\) 0 0
\(477\) −4.25219 −0.194694
\(478\) 0 0
\(479\) 4.24253 0.193846 0.0969231 0.995292i \(-0.469100\pi\)
0.0969231 + 0.995292i \(0.469100\pi\)
\(480\) 0 0
\(481\) 64.2150 2.92795
\(482\) 0 0
\(483\) −1.27841 −0.0581698
\(484\) 0 0
\(485\) 0.536960 0.0243821
\(486\) 0 0
\(487\) −5.96478 −0.270290 −0.135145 0.990826i \(-0.543150\pi\)
−0.135145 + 0.990826i \(0.543150\pi\)
\(488\) 0 0
\(489\) 5.35211 0.242031
\(490\) 0 0
\(491\) 6.20175 0.279881 0.139940 0.990160i \(-0.455309\pi\)
0.139940 + 0.990160i \(0.455309\pi\)
\(492\) 0 0
\(493\) 0.00504865 0.000227380 0
\(494\) 0 0
\(495\) −1.58722 −0.0713401
\(496\) 0 0
\(497\) −0.641953 −0.0287955
\(498\) 0 0
\(499\) 17.6753 0.791253 0.395626 0.918412i \(-0.370527\pi\)
0.395626 + 0.918412i \(0.370527\pi\)
\(500\) 0 0
\(501\) −17.3553 −0.775376
\(502\) 0 0
\(503\) −34.1635 −1.52328 −0.761638 0.648003i \(-0.775604\pi\)
−0.761638 + 0.648003i \(0.775604\pi\)
\(504\) 0 0
\(505\) 4.51488 0.200909
\(506\) 0 0
\(507\) 49.3512 2.19176
\(508\) 0 0
\(509\) −18.4383 −0.817265 −0.408633 0.912699i \(-0.633994\pi\)
−0.408633 + 0.912699i \(0.633994\pi\)
\(510\) 0 0
\(511\) 10.3124 0.456195
\(512\) 0 0
\(513\) 31.2679 1.38051
\(514\) 0 0
\(515\) −0.166531 −0.00733824
\(516\) 0 0
\(517\) 7.98478 0.351170
\(518\) 0 0
\(519\) 19.8529 0.871447
\(520\) 0 0
\(521\) −11.8840 −0.520647 −0.260323 0.965521i \(-0.583829\pi\)
−0.260323 + 0.965521i \(0.583829\pi\)
\(522\) 0 0
\(523\) −12.8711 −0.562815 −0.281407 0.959588i \(-0.590801\pi\)
−0.281407 + 0.959588i \(0.590801\pi\)
\(524\) 0 0
\(525\) −6.10096 −0.266268
\(526\) 0 0
\(527\) 2.31894 0.101015
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 10.7957 0.468495
\(532\) 0 0
\(533\) −76.2284 −3.30182
\(534\) 0 0
\(535\) 0.106448 0.00460214
\(536\) 0 0
\(537\) −26.2777 −1.13397
\(538\) 0 0
\(539\) −2.43556 −0.104907
\(540\) 0 0
\(541\) −30.3533 −1.30499 −0.652496 0.757792i \(-0.726278\pi\)
−0.652496 + 0.757792i \(0.726278\pi\)
\(542\) 0 0
\(543\) −7.75282 −0.332706
\(544\) 0 0
\(545\) −7.88629 −0.337811
\(546\) 0 0
\(547\) −34.2001 −1.46229 −0.731144 0.682223i \(-0.761013\pi\)
−0.731144 + 0.682223i \(0.761013\pi\)
\(548\) 0 0
\(549\) −14.2875 −0.609775
\(550\) 0 0
\(551\) −0.0934403 −0.00398069
\(552\) 0 0
\(553\) 2.43556 0.103571
\(554\) 0 0
\(555\) 5.45335 0.231482
\(556\) 0 0
\(557\) −29.2289 −1.23847 −0.619235 0.785206i \(-0.712557\pi\)
−0.619235 + 0.785206i \(0.712557\pi\)
\(558\) 0 0
\(559\) −36.1361 −1.52839
\(560\) 0 0
\(561\) −0.942515 −0.0397930
\(562\) 0 0
\(563\) 23.4170 0.986907 0.493454 0.869772i \(-0.335734\pi\)
0.493454 + 0.869772i \(0.335734\pi\)
\(564\) 0 0
\(565\) 7.64376 0.321575
\(566\) 0 0
\(567\) −3.03800 −0.127584
\(568\) 0 0
\(569\) −31.8255 −1.33419 −0.667097 0.744971i \(-0.732464\pi\)
−0.667097 + 0.744971i \(0.732464\pi\)
\(570\) 0 0
\(571\) 12.8639 0.538336 0.269168 0.963093i \(-0.413251\pi\)
0.269168 + 0.963093i \(0.413251\pi\)
\(572\) 0 0
\(573\) −18.2607 −0.762850
\(574\) 0 0
\(575\) 4.77229 0.199018
\(576\) 0 0
\(577\) −4.25457 −0.177120 −0.0885600 0.996071i \(-0.528226\pi\)
−0.0885600 + 0.996071i \(0.528226\pi\)
\(578\) 0 0
\(579\) 15.8570 0.658995
\(580\) 0 0
\(581\) −11.7343 −0.486820
\(582\) 0 0
\(583\) −7.58350 −0.314076
\(584\) 0 0
\(585\) −4.68141 −0.193552
\(586\) 0 0
\(587\) −27.9185 −1.15232 −0.576160 0.817337i \(-0.695450\pi\)
−0.576160 + 0.817337i \(0.695450\pi\)
\(588\) 0 0
\(589\) −42.9189 −1.76844
\(590\) 0 0
\(591\) −13.6048 −0.559627
\(592\) 0 0
\(593\) −18.8711 −0.774944 −0.387472 0.921881i \(-0.626652\pi\)
−0.387472 + 0.921881i \(0.626652\pi\)
\(594\) 0 0
\(595\) −0.144448 −0.00592181
\(596\) 0 0
\(597\) −14.2310 −0.582437
\(598\) 0 0
\(599\) −4.56833 −0.186657 −0.0933285 0.995635i \(-0.529751\pi\)
−0.0933285 + 0.995635i \(0.529751\pi\)
\(600\) 0 0
\(601\) 8.16113 0.332899 0.166450 0.986050i \(-0.446770\pi\)
0.166450 + 0.986050i \(0.446770\pi\)
\(602\) 0 0
\(603\) 3.80273 0.154859
\(604\) 0 0
\(605\) 2.41844 0.0983235
\(606\) 0 0
\(607\) 17.3331 0.703529 0.351764 0.936089i \(-0.385582\pi\)
0.351764 + 0.936089i \(0.385582\pi\)
\(608\) 0 0
\(609\) 0.0213220 0.000864012 0
\(610\) 0 0
\(611\) 23.5507 0.952758
\(612\) 0 0
\(613\) 11.5952 0.468325 0.234162 0.972198i \(-0.424765\pi\)
0.234162 + 0.972198i \(0.424765\pi\)
\(614\) 0 0
\(615\) −6.47356 −0.261039
\(616\) 0 0
\(617\) −12.6206 −0.508087 −0.254044 0.967193i \(-0.581761\pi\)
−0.254044 + 0.967193i \(0.581761\pi\)
\(618\) 0 0
\(619\) −35.3244 −1.41981 −0.709904 0.704298i \(-0.751262\pi\)
−0.709904 + 0.704298i \(0.751262\pi\)
\(620\) 0 0
\(621\) 5.58112 0.223963
\(622\) 0 0
\(623\) −7.59085 −0.304121
\(624\) 0 0
\(625\) 21.6361 0.865446
\(626\) 0 0
\(627\) 17.4441 0.696649
\(628\) 0 0
\(629\) −2.70592 −0.107892
\(630\) 0 0
\(631\) −10.5356 −0.419417 −0.209708 0.977764i \(-0.567251\pi\)
−0.209708 + 0.977764i \(0.567251\pi\)
\(632\) 0 0
\(633\) −32.4064 −1.28804
\(634\) 0 0
\(635\) 0.0333570 0.00132373
\(636\) 0 0
\(637\) −7.18356 −0.284623
\(638\) 0 0
\(639\) 0.876689 0.0346813
\(640\) 0 0
\(641\) 22.8906 0.904124 0.452062 0.891987i \(-0.350689\pi\)
0.452062 + 0.891987i \(0.350689\pi\)
\(642\) 0 0
\(643\) −22.8893 −0.902664 −0.451332 0.892356i \(-0.649051\pi\)
−0.451332 + 0.892356i \(0.649051\pi\)
\(644\) 0 0
\(645\) −3.06879 −0.120834
\(646\) 0 0
\(647\) −36.5209 −1.43578 −0.717892 0.696154i \(-0.754893\pi\)
−0.717892 + 0.696154i \(0.754893\pi\)
\(648\) 0 0
\(649\) 19.2535 0.755764
\(650\) 0 0
\(651\) 9.79361 0.383842
\(652\) 0 0
\(653\) −15.4493 −0.604577 −0.302288 0.953217i \(-0.597751\pi\)
−0.302288 + 0.953217i \(0.597751\pi\)
\(654\) 0 0
\(655\) 3.88891 0.151952
\(656\) 0 0
\(657\) −14.0833 −0.549441
\(658\) 0 0
\(659\) −46.1664 −1.79839 −0.899194 0.437550i \(-0.855846\pi\)
−0.899194 + 0.437550i \(0.855846\pi\)
\(660\) 0 0
\(661\) 38.6047 1.50155 0.750774 0.660559i \(-0.229681\pi\)
0.750774 + 0.660559i \(0.229681\pi\)
\(662\) 0 0
\(663\) −2.77990 −0.107962
\(664\) 0 0
\(665\) 2.67345 0.103672
\(666\) 0 0
\(667\) −0.0166785 −0.000645794 0
\(668\) 0 0
\(669\) 4.42204 0.170966
\(670\) 0 0
\(671\) −25.4808 −0.983674
\(672\) 0 0
\(673\) −16.7906 −0.647232 −0.323616 0.946189i \(-0.604899\pi\)
−0.323616 + 0.946189i \(0.604899\pi\)
\(674\) 0 0
\(675\) 26.6347 1.02517
\(676\) 0 0
\(677\) 28.7560 1.10518 0.552591 0.833452i \(-0.313639\pi\)
0.552591 + 0.833452i \(0.313639\pi\)
\(678\) 0 0
\(679\) −1.12525 −0.0431830
\(680\) 0 0
\(681\) −5.42371 −0.207837
\(682\) 0 0
\(683\) 10.1501 0.388384 0.194192 0.980964i \(-0.437791\pi\)
0.194192 + 0.980964i \(0.437791\pi\)
\(684\) 0 0
\(685\) 4.49896 0.171896
\(686\) 0 0
\(687\) −18.0228 −0.687612
\(688\) 0 0
\(689\) −22.3671 −0.852119
\(690\) 0 0
\(691\) −3.33903 −0.127023 −0.0635113 0.997981i \(-0.520230\pi\)
−0.0635113 + 0.997981i \(0.520230\pi\)
\(692\) 0 0
\(693\) 3.32615 0.126350
\(694\) 0 0
\(695\) 6.45211 0.244743
\(696\) 0 0
\(697\) 3.21214 0.121669
\(698\) 0 0
\(699\) −37.1045 −1.40342
\(700\) 0 0
\(701\) −35.3756 −1.33612 −0.668060 0.744107i \(-0.732875\pi\)
−0.668060 + 0.744107i \(0.732875\pi\)
\(702\) 0 0
\(703\) 50.0811 1.88885
\(704\) 0 0
\(705\) 2.00000 0.0753244
\(706\) 0 0
\(707\) −9.46131 −0.355829
\(708\) 0 0
\(709\) 41.7761 1.56894 0.784468 0.620170i \(-0.212936\pi\)
0.784468 + 0.620170i \(0.212936\pi\)
\(710\) 0 0
\(711\) −3.32615 −0.124740
\(712\) 0 0
\(713\) −7.66075 −0.286897
\(714\) 0 0
\(715\) −8.34898 −0.312234
\(716\) 0 0
\(717\) −24.9938 −0.933412
\(718\) 0 0
\(719\) 3.24913 0.121172 0.0605861 0.998163i \(-0.480703\pi\)
0.0605861 + 0.998163i \(0.480703\pi\)
\(720\) 0 0
\(721\) 0.348980 0.0129967
\(722\) 0 0
\(723\) 13.5552 0.504122
\(724\) 0 0
\(725\) −0.0795946 −0.00295607
\(726\) 0 0
\(727\) −9.63289 −0.357264 −0.178632 0.983916i \(-0.557167\pi\)
−0.178632 + 0.983916i \(0.557167\pi\)
\(728\) 0 0
\(729\) 25.5538 0.946437
\(730\) 0 0
\(731\) 1.52272 0.0563198
\(732\) 0 0
\(733\) 33.8899 1.25175 0.625876 0.779922i \(-0.284741\pi\)
0.625876 + 0.779922i \(0.284741\pi\)
\(734\) 0 0
\(735\) −0.610051 −0.0225021
\(736\) 0 0
\(737\) 6.78192 0.249815
\(738\) 0 0
\(739\) −34.3805 −1.26471 −0.632353 0.774680i \(-0.717911\pi\)
−0.632353 + 0.774680i \(0.717911\pi\)
\(740\) 0 0
\(741\) 51.4503 1.89007
\(742\) 0 0
\(743\) −24.2805 −0.890766 −0.445383 0.895340i \(-0.646933\pi\)
−0.445383 + 0.895340i \(0.646933\pi\)
\(744\) 0 0
\(745\) −6.87908 −0.252030
\(746\) 0 0
\(747\) 16.0250 0.586325
\(748\) 0 0
\(749\) −0.223070 −0.00815082
\(750\) 0 0
\(751\) 22.6514 0.826560 0.413280 0.910604i \(-0.364383\pi\)
0.413280 + 0.910604i \(0.364383\pi\)
\(752\) 0 0
\(753\) 38.9615 1.41984
\(754\) 0 0
\(755\) −0.199888 −0.00727468
\(756\) 0 0
\(757\) −22.0528 −0.801523 −0.400762 0.916182i \(-0.631254\pi\)
−0.400762 + 0.916182i \(0.631254\pi\)
\(758\) 0 0
\(759\) 3.11365 0.113018
\(760\) 0 0
\(761\) 15.9416 0.577883 0.288941 0.957347i \(-0.406697\pi\)
0.288941 + 0.957347i \(0.406697\pi\)
\(762\) 0 0
\(763\) 16.5264 0.598295
\(764\) 0 0
\(765\) 0.197267 0.00713221
\(766\) 0 0
\(767\) 56.7870 2.05046
\(768\) 0 0
\(769\) 39.4438 1.42238 0.711190 0.703000i \(-0.248157\pi\)
0.711190 + 0.703000i \(0.248157\pi\)
\(770\) 0 0
\(771\) 17.4298 0.627719
\(772\) 0 0
\(773\) −42.0375 −1.51198 −0.755992 0.654581i \(-0.772845\pi\)
−0.755992 + 0.654581i \(0.772845\pi\)
\(774\) 0 0
\(775\) −36.5593 −1.31325
\(776\) 0 0
\(777\) −11.4279 −0.409975
\(778\) 0 0
\(779\) −59.4503 −2.13003
\(780\) 0 0
\(781\) 1.56352 0.0559470
\(782\) 0 0
\(783\) −0.0930847 −0.00332658
\(784\) 0 0
\(785\) −11.6223 −0.414818
\(786\) 0 0
\(787\) −6.98472 −0.248978 −0.124489 0.992221i \(-0.539729\pi\)
−0.124489 + 0.992221i \(0.539729\pi\)
\(788\) 0 0
\(789\) −12.3245 −0.438763
\(790\) 0 0
\(791\) −16.0181 −0.569539
\(792\) 0 0
\(793\) −75.1542 −2.66880
\(794\) 0 0
\(795\) −1.89949 −0.0673679
\(796\) 0 0
\(797\) 26.2541 0.929969 0.464984 0.885319i \(-0.346060\pi\)
0.464984 + 0.885319i \(0.346060\pi\)
\(798\) 0 0
\(799\) −0.992388 −0.0351082
\(800\) 0 0
\(801\) 10.3665 0.366283
\(802\) 0 0
\(803\) −25.1166 −0.886344
\(804\) 0 0
\(805\) 0.477194 0.0168189
\(806\) 0 0
\(807\) 32.8253 1.15550
\(808\) 0 0
\(809\) 3.30337 0.116140 0.0580701 0.998313i \(-0.481505\pi\)
0.0580701 + 0.998313i \(0.481505\pi\)
\(810\) 0 0
\(811\) −29.5817 −1.03875 −0.519377 0.854545i \(-0.673836\pi\)
−0.519377 + 0.854545i \(0.673836\pi\)
\(812\) 0 0
\(813\) 26.1820 0.918244
\(814\) 0 0
\(815\) −1.99778 −0.0699793
\(816\) 0 0
\(817\) −28.1824 −0.985979
\(818\) 0 0
\(819\) 9.81029 0.342799
\(820\) 0 0
\(821\) −54.4549 −1.90049 −0.950245 0.311504i \(-0.899167\pi\)
−0.950245 + 0.311504i \(0.899167\pi\)
\(822\) 0 0
\(823\) −12.7484 −0.444381 −0.222191 0.975003i \(-0.571321\pi\)
−0.222191 + 0.975003i \(0.571321\pi\)
\(824\) 0 0
\(825\) 14.8592 0.517333
\(826\) 0 0
\(827\) 22.2940 0.775239 0.387620 0.921819i \(-0.373297\pi\)
0.387620 + 0.921819i \(0.373297\pi\)
\(828\) 0 0
\(829\) −2.25845 −0.0784392 −0.0392196 0.999231i \(-0.512487\pi\)
−0.0392196 + 0.999231i \(0.512487\pi\)
\(830\) 0 0
\(831\) 2.11255 0.0732834
\(832\) 0 0
\(833\) 0.302704 0.0104881
\(834\) 0 0
\(835\) 6.47821 0.224188
\(836\) 0 0
\(837\) −42.7555 −1.47785
\(838\) 0 0
\(839\) 38.8393 1.34088 0.670442 0.741962i \(-0.266105\pi\)
0.670442 + 0.741962i \(0.266105\pi\)
\(840\) 0 0
\(841\) −28.9997 −0.999990
\(842\) 0 0
\(843\) −6.84200 −0.235651
\(844\) 0 0
\(845\) −18.4213 −0.633714
\(846\) 0 0
\(847\) −5.06804 −0.174140
\(848\) 0 0
\(849\) −0.934926 −0.0320866
\(850\) 0 0
\(851\) 8.93916 0.306431
\(852\) 0 0
\(853\) 3.34045 0.114375 0.0571873 0.998363i \(-0.481787\pi\)
0.0571873 + 0.998363i \(0.481787\pi\)
\(854\) 0 0
\(855\) −3.65102 −0.124862
\(856\) 0 0
\(857\) 35.3153 1.20635 0.603174 0.797610i \(-0.293903\pi\)
0.603174 + 0.797610i \(0.293903\pi\)
\(858\) 0 0
\(859\) 37.5433 1.28096 0.640481 0.767974i \(-0.278735\pi\)
0.640481 + 0.767974i \(0.278735\pi\)
\(860\) 0 0
\(861\) 13.5659 0.462324
\(862\) 0 0
\(863\) −32.1921 −1.09583 −0.547916 0.836534i \(-0.684579\pi\)
−0.547916 + 0.836534i \(0.684579\pi\)
\(864\) 0 0
\(865\) −7.41051 −0.251965
\(866\) 0 0
\(867\) −21.6159 −0.734114
\(868\) 0 0
\(869\) −5.93196 −0.201228
\(870\) 0 0
\(871\) 20.0029 0.677773
\(872\) 0 0
\(873\) 1.53670 0.0520095
\(874\) 0 0
\(875\) 4.66328 0.157647
\(876\) 0 0
\(877\) −9.78157 −0.330300 −0.165150 0.986268i \(-0.552811\pi\)
−0.165150 + 0.986268i \(0.552811\pi\)
\(878\) 0 0
\(879\) −13.1073 −0.442099
\(880\) 0 0
\(881\) 43.3634 1.46095 0.730475 0.682939i \(-0.239299\pi\)
0.730475 + 0.682939i \(0.239299\pi\)
\(882\) 0 0
\(883\) 46.7422 1.57300 0.786500 0.617590i \(-0.211891\pi\)
0.786500 + 0.617590i \(0.211891\pi\)
\(884\) 0 0
\(885\) 4.82254 0.162108
\(886\) 0 0
\(887\) −38.5913 −1.29577 −0.647885 0.761738i \(-0.724346\pi\)
−0.647885 + 0.761738i \(0.724346\pi\)
\(888\) 0 0
\(889\) −0.0699024 −0.00234445
\(890\) 0 0
\(891\) 7.39924 0.247884
\(892\) 0 0
\(893\) 18.3671 0.614632
\(894\) 0 0
\(895\) 9.80870 0.327869
\(896\) 0 0
\(897\) 9.18356 0.306630
\(898\) 0 0
\(899\) 0.127770 0.00426136
\(900\) 0 0
\(901\) 0.942515 0.0313997
\(902\) 0 0
\(903\) 6.43092 0.214008
\(904\) 0 0
\(905\) 2.89390 0.0961965
\(906\) 0 0
\(907\) −18.5641 −0.616411 −0.308205 0.951320i \(-0.599728\pi\)
−0.308205 + 0.951320i \(0.599728\pi\)
\(908\) 0 0
\(909\) 12.9209 0.428560
\(910\) 0 0
\(911\) −4.69995 −0.155716 −0.0778581 0.996964i \(-0.524808\pi\)
−0.0778581 + 0.996964i \(0.524808\pi\)
\(912\) 0 0
\(913\) 28.5796 0.945846
\(914\) 0 0
\(915\) −6.38234 −0.210994
\(916\) 0 0
\(917\) −8.14954 −0.269121
\(918\) 0 0
\(919\) −43.7947 −1.44465 −0.722326 0.691552i \(-0.756927\pi\)
−0.722326 + 0.691552i \(0.756927\pi\)
\(920\) 0 0
\(921\) −21.7127 −0.715458
\(922\) 0 0
\(923\) 4.61151 0.151790
\(924\) 0 0
\(925\) 42.6602 1.40266
\(926\) 0 0
\(927\) −0.476588 −0.0156532
\(928\) 0 0
\(929\) 27.4615 0.900982 0.450491 0.892781i \(-0.351249\pi\)
0.450491 + 0.892781i \(0.351249\pi\)
\(930\) 0 0
\(931\) −5.60244 −0.183613
\(932\) 0 0
\(933\) 10.3019 0.337268
\(934\) 0 0
\(935\) 0.351813 0.0115055
\(936\) 0 0
\(937\) −26.5337 −0.866819 −0.433410 0.901197i \(-0.642690\pi\)
−0.433410 + 0.901197i \(0.642690\pi\)
\(938\) 0 0
\(939\) −4.71480 −0.153862
\(940\) 0 0
\(941\) 54.5366 1.77784 0.888922 0.458059i \(-0.151455\pi\)
0.888922 + 0.458059i \(0.151455\pi\)
\(942\) 0 0
\(943\) −10.6115 −0.345558
\(944\) 0 0
\(945\) 2.66328 0.0866363
\(946\) 0 0
\(947\) −16.3392 −0.530951 −0.265476 0.964118i \(-0.585529\pi\)
−0.265476 + 0.964118i \(0.585529\pi\)
\(948\) 0 0
\(949\) −74.0800 −2.40474
\(950\) 0 0
\(951\) −5.50952 −0.178658
\(952\) 0 0
\(953\) −5.02906 −0.162907 −0.0814536 0.996677i \(-0.525956\pi\)
−0.0814536 + 0.996677i \(0.525956\pi\)
\(954\) 0 0
\(955\) 6.81616 0.220566
\(956\) 0 0
\(957\) −0.0519311 −0.00167869
\(958\) 0 0
\(959\) −9.42795 −0.304444
\(960\) 0 0
\(961\) 27.6871 0.893132
\(962\) 0 0
\(963\) 0.304638 0.00981683
\(964\) 0 0
\(965\) −5.91895 −0.190538
\(966\) 0 0
\(967\) 55.4970 1.78466 0.892332 0.451379i \(-0.149068\pi\)
0.892332 + 0.451379i \(0.149068\pi\)
\(968\) 0 0
\(969\) −2.16804 −0.0696473
\(970\) 0 0
\(971\) 53.3518 1.71214 0.856071 0.516858i \(-0.172899\pi\)
0.856071 + 0.516858i \(0.172899\pi\)
\(972\) 0 0
\(973\) −13.5209 −0.433462
\(974\) 0 0
\(975\) 43.8266 1.40357
\(976\) 0 0
\(977\) 10.0073 0.320161 0.160080 0.987104i \(-0.448825\pi\)
0.160080 + 0.987104i \(0.448825\pi\)
\(978\) 0 0
\(979\) 18.4880 0.590878
\(980\) 0 0
\(981\) −22.5694 −0.720585
\(982\) 0 0
\(983\) −16.7241 −0.533414 −0.266707 0.963778i \(-0.585936\pi\)
−0.266707 + 0.963778i \(0.585936\pi\)
\(984\) 0 0
\(985\) 5.07827 0.161807
\(986\) 0 0
\(987\) −4.19117 −0.133406
\(988\) 0 0
\(989\) −5.03039 −0.159957
\(990\) 0 0
\(991\) 8.96165 0.284676 0.142338 0.989818i \(-0.454538\pi\)
0.142338 + 0.989818i \(0.454538\pi\)
\(992\) 0 0
\(993\) 34.5536 1.09653
\(994\) 0 0
\(995\) 5.31202 0.168402
\(996\) 0 0
\(997\) 11.7263 0.371376 0.185688 0.982609i \(-0.440549\pi\)
0.185688 + 0.982609i \(0.440549\pi\)
\(998\) 0 0
\(999\) 49.8905 1.57847
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1288.2.a.n.1.4 4
4.3 odd 2 2576.2.a.ba.1.1 4
7.6 odd 2 9016.2.a.bf.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1288.2.a.n.1.4 4 1.1 even 1 trivial
2576.2.a.ba.1.1 4 4.3 odd 2
9016.2.a.bf.1.1 4 7.6 odd 2